You don't create carbon out of thin air, it's from the fuel, so burning the same quantity of fuel will result in the same quantity of carbon, no matter how the engine works. Therefore a tax on fuel is a tax on carbon.
Yes but since incomplete combustion is inverse correlated with fuel efficiency (unburned fuel is wasted fuel), it's not really a trade off. What is a trade off is NO emissions vs fuel efficiency. Burning your fuel oxygen rich will burn of more fuel, but also makes more NO (due to higher temperatures if I remember correctly).
By definition, more carbon is less efficiency. Efficiency is about how much of the hydrocarbon you turn into heat. Diesels often burn a little dirty. That's partly because diesel engines don't burn all the fuel
